The phrase at least two dice are less than or equal to 4 means that either only two of the dice are lower than or equal to 4 or that all three dice are lower than or equal to 4.
Probability: ≈ 0.74 Explanation: See solution.
Practice makes perfect
We are asked to find the probability that after rolling 3 dice at least two of them will be less than or equal to 4.
P(at least two dice≤ 4)=?
The phrase at least two means two or more. In other words, either only two of the three dice are less than or equal to 4, or all three dice are less than or equal to 4. These events are mutually exclusive, and thus we can write the following equation.
P(at least two dice≤ 4)=
P(only two dice≤ 4)+P(all three dice≤ 4)
Let's calculate these probabilities one at a time. We can start with calculating P(only two dice ≤ 4).
P(only two dice≤ 4)
We will find the probability that any two of the dice are less than or equal to 4 and the other die is greater than 4.
After rolling a six sided die, there are 6 possible outcomes. 4 of them are less than or equal to 4. We can calculate the probability that a die is less than or equal to 4 using this information.
P(die ≤ 4)=4/6=2/3
Similarly, there are 2 rolls that are higher than 4. Let's calculate the probability that a die is greater than 4.
P(die>4)=2/6=1/3
Next, notice that the events of rolling the dice do not affect each other. Thus, the events are independent. We can find the probability that the first two dice are less than or equal to 4 and the other die is greater than 4 using the formula for the probability of independent events.
Probability=
P(die≤ 4)* P(die≤ 4)* P(die>4)
Let's substitute the probabilities.
The probability that the first two dice are less than or equal to 4 and that the third die is greater than 4 is 427. Notice that there are other orders of the dice in which only two of the dice are less than or equal to 4.
Possible Orders of the Dice
1. First and second dice lower than four, third die higher than four.
2. First and third dice lower than four, second die higher than four.
3. Second and third dice lower than four, first die higher than four.
Since there are 3 possible orders we need to multiply our result, 427, by 3 to obtain P(only two dice≤ 4).
P(only two dice≤ 4)= 3* 4/27≈ 0.44
Now, let's find the probability that all three dice are less than or equal to 4.
P(all three dice ≤ 4)
In the previous part, we found that the probability that a die is less than or equal to 4 is 23.
P(die ≤ 4)=2/3
Also, we concluded that the rolls are independent. Thus, we can use the formula for the probability of independent events to calculate the probability that all three dice are less than or equal to 4.
P(all three dice≤ 4)= P(die ≤ 4)* P(die ≤ 4) * P(die ≤ 4)
Let's substitute!
Finally, we are ready to calculate P(at least two dice ≤ 4).
P(at least two dice≤ 4)
We found that P(only two dice ≤ 4) is about 0.44 and that P(all three dice ≤ 4) is about 0.3. Let's substitute these values into the equation for P(at least two dice ≤ 4).
P(at least two dice≤ 4)=
P(only two dice≤ 4)+P(all three dice≤ 4)
⇕
P(at least two dice≤ 4)≈ 0.44+ 0.3=0.74
The probability is equal to about 0.74.
